Closed form for $\int_0^\infty\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x\phantom{|}}\sqrt{x^2+1}}e^{-x}dx$

Solution 1:

I introduce a parameter $a$, $$ I(a)=\int_0^\infty\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x\phantom{|}}\sqrt{x^2+1}}e^{-ax}dx $$ then took a Mellin transform from $a$ to $s$, $$ \mathcal{M}_{a\to s}[I(a)]=\Gamma(s)\int_0^\infty x^{-s-\frac{1}{2}}\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x^2+1}}\; dx $$ which apparently equals $$ \mathcal{M}_{a\to s}[I(a)]=\sqrt{2\pi} \frac{\Gamma(s)^2}{\Gamma(s+\frac{1}{2})}\cos\left(\frac{\pi s}{2}\right)\sec(\pi s) $$ then the inverse Mellin transform gives $$ I(a)= \frac{\pi}{\sqrt{2}}\left(\sin \left(\frac a 2 \right)J_0 \left(\frac a 2 \right)-\cos\left(\frac a 2 \right)Y_0\left(\frac a 2 \right) \right) $$